Integrand size = 25, antiderivative size = 776 \[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=-\frac {\arctan \left (\frac {\sqrt {-a} \cos (c+d x)}{\sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}\right )}{4 \sqrt {-a} d}-\frac {\sqrt {b} \cos (c+d x) \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}{2 a \sqrt {a+b} d \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )}-\frac {\sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)} \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\sqrt [4]{b} (a+b)^{3/4} \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{(a+b) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{2 a d \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}-\frac {\sqrt [4]{b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{(a+b) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{2 a \sqrt [4]{a+b} d \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}-\frac {\sqrt [4]{a+b} \left (\sqrt {b}-\sqrt {a+b}\right )^2 \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{(a+b) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b}+\sqrt {a+b}\right )^2}{4 \sqrt {b} \sqrt {a+b}},2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{8 a \sqrt [4]{b} d \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}} \]
-1/4*arctan(cos(d*x+c)*(-a)^(1/2)/(a+b-2*b*cos(d*x+c)^2+b*cos(d*x+c)^4)^(1 /2))/d/(-a)^(1/2)-1/2*cot(d*x+c)*csc(d*x+c)*(a+b-2*b*cos(d*x+c)^2+b*cos(d* x+c)^4)^(1/2)/a/d-1/2*cos(d*x+c)*b^(1/2)*(a+b-2*b*cos(d*x+c)^2+b*cos(d*x+c )^4)^(1/2)/a/d/(1+cos(d*x+c)^2*b^(1/2)/(a+b)^(1/2))/(a+b)^(1/2)+1/2*b^(1/4 )*(a+b)^(3/4)*(cos(2*arctan(b^(1/4)*cos(d*x+c)/(a+b)^(1/4)))^2)^(1/2)/cos( 2*arctan(b^(1/4)*cos(d*x+c)/(a+b)^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*c os(d*x+c)/(a+b)^(1/4))),1/2*(2+2*b^(1/2)/(a+b)^(1/2))^(1/2))*(1+cos(d*x+c) ^2*b^(1/2)/(a+b)^(1/2))*((a+b-2*b*cos(d*x+c)^2+b*cos(d*x+c)^4)/(a+b)/(1+co s(d*x+c)^2*b^(1/2)/(a+b)^(1/2))^2)^(1/2)/a/d/(a+b-2*b*cos(d*x+c)^2+b*cos(d *x+c)^4)^(1/2)-1/8*(a+b)^(1/4)*(cos(2*arctan(b^(1/4)*cos(d*x+c)/(a+b)^(1/4 )))^2)^(1/2)/cos(2*arctan(b^(1/4)*cos(d*x+c)/(a+b)^(1/4)))*EllipticPi(sin( 2*arctan(b^(1/4)*cos(d*x+c)/(a+b)^(1/4))),1/4*(b^(1/2)+(a+b)^(1/2))^2/b^(1 /2)/(a+b)^(1/2),1/2*(2+2*b^(1/2)/(a+b)^(1/2))^(1/2))*(1+cos(d*x+c)^2*b^(1/ 2)/(a+b)^(1/2))*(b^(1/2)-(a+b)^(1/2))^2*((a+b-2*b*cos(d*x+c)^2+b*cos(d*x+c )^4)/(a+b)/(1+cos(d*x+c)^2*b^(1/2)/(a+b)^(1/2))^2)^(1/2)/a/b^(1/4)/d/(a+b- 2*b*cos(d*x+c)^2+b*cos(d*x+c)^4)^(1/2)-1/2*b^(1/4)*(cos(2*arctan(b^(1/4)*c os(d*x+c)/(a+b)^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*cos(d*x+c)/(a+b)^(1/ 4)))*EllipticF(sin(2*arctan(b^(1/4)*cos(d*x+c)/(a+b)^(1/4))),1/2*(2+2*b^(1 /2)/(a+b)^(1/2))^(1/2))*(1+cos(d*x+c)^2*b^(1/2)/(a+b)^(1/2))*(a+b-b^(1/2)* (a+b)^(1/2))*((a+b-2*b*cos(d*x+c)^2+b*cos(d*x+c)^4)/(a+b)/(1+cos(d*x+c)...
Result contains complex when optimal does not.
Time = 25.16 (sec) , antiderivative size = 1442, normalized size of antiderivative = 1.86 \[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx =\text {Too large to display} \]
-1/4*(Sqrt[8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)]]*Cot[c + d*x]*Csc[c + d*x])/(Sqrt[2]*a*d) + (Sec[c + d*x]*(-a - 2*a*Tan[c + d*x]^2 - a*Tan[c + d*x]^4 - b*Tan[c + d*x]^4 - (b*(Sqrt[b]*EllipticE[ArcSin[Sqrt[ (I*(a - I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[c + d*x]^2))/(Sqrt[a] *Sqrt[b])]/Sqrt[2]], (2*Sqrt[a])/(Sqrt[a] - I*Sqrt[b])] + I*(Sqrt[a] + I*S qrt[b])*EllipticF[ArcSin[Sqrt[(I*(a - I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[c + d*x]^2))/(Sqrt[a]*Sqrt[b])]/Sqrt[2]], (2*Sqrt[a])/(Sqrt[a] - I*Sqrt[b])])*Sqrt[(((-I)*Sqrt[a] + Sqrt[b])*(1 + Tan[c + d*x]^2))/Sqrt[b]] *Sqrt[(I*(a - I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[c + d*x]^2))/(S qrt[a]*Sqrt[b])]*(Sqrt[b]*Tan[c + d*x]^2 + I*Sqrt[a]*(1 + Tan[c + d*x]^2)) )/((a + b)*Sqrt[((-I)*(a + I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[c + d*x]^2))/(Sqrt[a]*Sqrt[b])]) + (a*((-I)*Sqrt[b]*EllipticE[ArcSin[Sqrt[(I *(a - I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[c + d*x]^2))/(Sqrt[a]*S qrt[b])]/Sqrt[2]], (2*Sqrt[a])/(Sqrt[a] - I*Sqrt[b])] + (Sqrt[a] + I*Sqrt[ b])*EllipticF[ArcSin[Sqrt[(I*(a - I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b *Tan[c + d*x]^2))/(Sqrt[a]*Sqrt[b])]/Sqrt[2]], (2*Sqrt[a])/(Sqrt[a] - I*Sq rt[b])])*Sqrt[(((-I)*Sqrt[a] + Sqrt[b])*(1 + Tan[c + d*x]^2))/Sqrt[b]]*Sqr t[(I*(a - I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[c + d*x]^2))/(Sqrt[ a]*Sqrt[b])]*((-I)*Sqrt[b]*Tan[c + d*x]^2 + Sqrt[a]*(1 + Tan[c + d*x]^2))) /((a + b)*Sqrt[((-I)*(a + I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[...
Time = 1.30 (sec) , antiderivative size = 802, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3042, 3694, 1551, 25, 2232, 25, 27, 1509, 2226, 1416, 2222}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sin (c+d x)^3 \sqrt {a+b \sin (c+d x)^4}}dx\) |
\(\Big \downarrow \) 3694 |
\(\displaystyle -\frac {\int \frac {1}{\left (1-\cos ^2(c+d x)\right )^2 \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)}{d}\) |
\(\Big \downarrow \) 1551 |
\(\displaystyle -\frac {\frac {\cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a \left (1-\cos ^2(c+d x)\right )}-\frac {\int -\frac {-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b}{\left (1-\cos ^2(c+d x)\right ) \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)}{2 a}}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\frac {\int \frac {-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b}{\left (1-\cos ^2(c+d x)\right ) \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)}{2 a}+\frac {\cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a \left (1-\cos ^2(c+d x)\right )}}{d}\) |
\(\Big \downarrow \) 2232 |
\(\displaystyle -\frac {\frac {-\sqrt {b} \sqrt {a+b} \int \frac {1-\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}}{\sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)-\frac {\int -\frac {b \left (\sqrt {b} \left (\sqrt {b}-\sqrt {a+b}\right ) \cos ^2(c+d x)+a-b+\sqrt {b} \sqrt {a+b}\right )}{\left (1-\cos ^2(c+d x)\right ) \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)}{b}}{2 a}+\frac {\cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a \left (1-\cos ^2(c+d x)\right )}}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\frac {\frac {\int \frac {b \left (\sqrt {b} \left (\sqrt {b}-\sqrt {a+b}\right ) \cos ^2(c+d x)+a-b+\sqrt {b} \sqrt {a+b}\right )}{\left (1-\cos ^2(c+d x)\right ) \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)}{b}-\sqrt {b} \sqrt {a+b} \int \frac {1-\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}}{\sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)}{2 a}+\frac {\cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a \left (1-\cos ^2(c+d x)\right )}}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {\int \frac {\sqrt {b} \left (\sqrt {b}-\sqrt {a+b}\right ) \cos ^2(c+d x)+a-b+\sqrt {b} \sqrt {a+b}}{\left (1-\cos ^2(c+d x)\right ) \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)-\sqrt {b} \sqrt {a+b} \int \frac {1-\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}}{\sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)}{2 a}+\frac {\cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a \left (1-\cos ^2(c+d x)\right )}}{d}\) |
\(\Big \downarrow \) 1509 |
\(\displaystyle -\frac {\frac {\int \frac {\sqrt {b} \left (\sqrt {b}-\sqrt {a+b}\right ) \cos ^2(c+d x)+a-b+\sqrt {b} \sqrt {a+b}}{\left (1-\cos ^2(c+d x)\right ) \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)-\sqrt {b} \sqrt {a+b} \left (\frac {\sqrt [4]{a+b} \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{\sqrt [4]{b} \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac {\cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )}\right )}{2 a}+\frac {\cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a \left (1-\cos ^2(c+d x)\right )}}{d}\) |
\(\Big \downarrow \) 2226 |
\(\displaystyle -\frac {\frac {\frac {2 \sqrt {b} \left (-\sqrt {b} \sqrt {a+b}+a+b\right ) \int \frac {1}{\sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)}{\sqrt {a+b}}-\sqrt {a+b} \left (\sqrt {b}-\sqrt {a+b}\right ) \int \frac {\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1}{\left (1-\cos ^2(c+d x)\right ) \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)-\sqrt {b} \sqrt {a+b} \left (\frac {\sqrt [4]{a+b} \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{\sqrt [4]{b} \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac {\cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )}\right )}{2 a}+\frac {\cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a \left (1-\cos ^2(c+d x)\right )}}{d}\) |
\(\Big \downarrow \) 1416 |
\(\displaystyle -\frac {\frac {-\sqrt {a+b} \left (\sqrt {b}-\sqrt {a+b}\right ) \int \frac {\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1}{\left (1-\cos ^2(c+d x)\right ) \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)+\frac {\sqrt [4]{b} \left (-\sqrt {b} \sqrt {a+b}+a+b\right ) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{\sqrt [4]{a+b} \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\sqrt {b} \sqrt {a+b} \left (\frac {\sqrt [4]{a+b} \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{\sqrt [4]{b} \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac {\cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )}\right )}{2 a}+\frac {\cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a \left (1-\cos ^2(c+d x)\right )}}{d}\) |
\(\Big \downarrow \) 2222 |
\(\displaystyle -\frac {\frac {\sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b} \cos (c+d x)}{2 a \left (1-\cos ^2(c+d x)\right )}+\frac {-\sqrt {b} \sqrt {a+b} \left (\frac {\sqrt [4]{a+b} \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{\sqrt [4]{b} \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}-\frac {\cos (c+d x) \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )}\right )+\frac {\sqrt [4]{b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{\sqrt [4]{a+b} \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}-\sqrt {a+b} \left (\sqrt {b}-\sqrt {a+b}\right ) \left (\frac {\left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}\right )}{2 \sqrt {a}}+\frac {\sqrt [4]{a+b} \left (1-\frac {\sqrt {b}}{\sqrt {a+b}}\right ) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b}+\sqrt {a+b}\right )^2}{4 \sqrt {b} \sqrt {a+b}},2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{4 \sqrt [4]{b} \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}\right )}{2 a}}{d}\) |
-(((Cos[c + d*x]*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])/(2*a *(1 - Cos[c + d*x]^2)) + (-(Sqrt[b]*Sqrt[a + b]*(-((Cos[c + d*x]*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b]))) + ((a + b)^(1/4)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqr t[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticE[2*ArcTan[(b^(1/4)*C os[c + d*x])/(a + b)^(1/4)], (1 + Sqrt[b]/Sqrt[a + b])/2])/(b^(1/4)*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]))) + (b^(1/4)*(a + b - Sqrt[ b]*Sqrt[a + b])*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2 *b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^ 2)/Sqrt[a + b])^2)]*EllipticF[2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4 )], (1 + Sqrt[b]/Sqrt[a + b])/2])/((a + b)^(1/4)*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]) - Sqrt[a + b]*(Sqrt[b] - Sqrt[a + b])*(((1 + S qrt[b]/Sqrt[a + b])*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]])/(2*Sqrt[a]) + ((a + b)^(1/4)*(1 - Sqrt[b]/S qrt[a + b])*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*C os[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/S qrt[a + b])^2)]*EllipticPi[(Sqrt[b] + Sqrt[a + b])^2/(4*Sqrt[b]*Sqrt[a + b ]), 2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1 + Sqrt[b]/Sqrt[a + b])/2])/(4*b^(1/4)*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])...
3.3.45.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)^(q_)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_ Symbol] :> Simp[(-e^2)*x*(d + e*x^2)^(q + 1)*(Sqrt[a + b*x^2 + c*x^4]/(2*d* (q + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/(2*d*(q + 1)*(c*d^2 - b*d*e + a*e^2)) Int[((d + e*x^2)^(q + 1)/Sqrt[a + b*x^2 + c*x^4])*Simp[a*e^2*(2 *q + 3) + 2*d*(c*d - b*e)*(q + 1) - 2*e*(c*d*(q + 1) - b*e*(q + 2))*x^2 + c *e^2*(2*q + 5)*x^4, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4* a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[q, -1]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A rcTanh[Rt[b - c*(d/e) - a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ b - c*(d/e) - a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*Ell ipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[-b + c*(d/e) + a*(e/d)]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[(A*(c*d + a*e*q) - a*B*(e + d*q))/(c*d^2 - a*e^2) Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Simp[a*(B*d - A*e)*((e + d*q)/(c*d^2 - a*e^2)) Int[(1 + q*x^2)/((d + e*x^ 2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && NeQ[c*A^2 - a*B^2, 0]
Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]) , x_Symbol] :> With[{q = Rt[c/a, 2], A = Coeff[P4x, x, 0], B = Coeff[P4x, x , 2], C = Coeff[P4x, x, 4]}, Simp[-C/(e*q) Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] + Simp[1/(c*e) Int[(A*c*e + a*C*d*q + (B*c*e - C*(c*d - a*e*q))*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b , c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && !GtQ[b^2 - 4*a*c, 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
\[\int \frac {\csc ^{3}\left (d x +c \right )}{\sqrt {a +b \left (\sin ^{4}\left (d x +c \right )\right )}}d x\]
\[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int { \frac {\csc \left (d x + c\right )^{3}}{\sqrt {b \sin \left (d x + c\right )^{4} + a}} \,d x } \]
\[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int \frac {\csc ^{3}{\left (c + d x \right )}}{\sqrt {a + b \sin ^{4}{\left (c + d x \right )}}}\, dx \]
\[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int { \frac {\csc \left (d x + c\right )^{3}}{\sqrt {b \sin \left (d x + c\right )^{4} + a}} \,d x } \]
\[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int { \frac {\csc \left (d x + c\right )^{3}}{\sqrt {b \sin \left (d x + c\right )^{4} + a}} \,d x } \]
Timed out. \[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int \frac {1}{{\sin \left (c+d\,x\right )}^3\,\sqrt {b\,{\sin \left (c+d\,x\right )}^4+a}} \,d x \]